Background, Motivation and Objective Ultrasound NDE data is comprised of a sequence of echoes backscattered from the reflectors situated along the transducers focused beam. These backscattered echoes contain valuable information pertaining to the acoustic impedance, geometric shape, and size of the reflectors. The ultimate objective of ultrasound imaging systems is to display this information termed as reflectivity function (RF). However, the backscattered echoes convey only the convolved version of the RF by the bandlimited transducer pulse. Various deconvolution techniques have been proposed over the years to restore the RF by inverting this convolution process. More recently, a new class of deconvolution algorithms is emerged in the compressive sensing (CS) field to exploit the sparsity of RF. In contrast to classical inversion based methods, these sparse deconvolution methods (SDMs) directly estimate the RF and provide significant resolution gains. However, SDMs generally assume the ultrasound pulse is invariant throughout the propagation path. While convenient, this assumption is not realistic in ultrasound NDE due to the frequency dependent absorption and scattering. As such, pulse variance severely compromises the performance of SDMs and leads to inaccurate estimations of the RF. The goal of this study is to develop a restoration technique that is sparse while accounting for pulse variance.